How do you exchange an object/mesh between two separate rigged armatures
To hold and drop an object, here is the basic operation to do:


Give the object a Child Of constraint, press Set Inverse so that it stays in place.





Make the parent move from point A (at frame 0) to point B (let's say at frame 20).





The frame before the frame where the dropping is supposed to happen (let say frame 9 as the dropping will happen frame 10), select the child, create a location keyframe and keyframe the Child Of constraint Influence at 1.





The next frame (so, here, frame 10), where the dropping happens, keep the child selected, create a keyframe Visual LocRotScale (or simply Visual Location), so that the new location is the one currently seen on the 3D view. Also keyframe the Child Of constraint Influence at 0 so that the object is not the child anymore.





Now as you give the object to another one, just give this object a second Child Of constraint with the new parent. At frame 9 you'll keyframe the Influence at 0, at frame 10 you'll keyframe the Influence at 1, and now the the child has a new parent.